#495289

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

627 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,23,30,37,19,24,31,38,32,39,42,44)(15,20,25,33,26,34,40,16,21,28,35,29,36,41,43,27);(6,9)(10,12)(13,15,17,25,18,26,30,40,19,21,31,35,32,36,42,43)(14,20,22,33,23,34,37,16,24,28,38,29,39,41,44,27);(1,2,3,4)(13,16)(14,21)(15,24)(17,28)(18,29)(19,27)(20,31)(22,35)(23,36)(25,38)(26,39)(30,41)(32,33)(34,42)(37,43)(40,44);(5,7,8,11)(6,10,9,12)(15,21)(16,27)(20,28)(25,35)(26,36)(29,33)(34,41)(40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=di,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402485);